arXiv subjects

Stanley Snelson

Publications and source records attributed to Stanley Snelson.

26 records · Page 2Linked to original sources

arXiv2019

Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art.

math.AP↗

arXiv2018

Local existence, lower mass bounds, and a new continuation criterion for the Landau equation

We consider the spatially inhomogeneous Landau equation with soft potentials, including the case of Coulomb interactions. First, we establish the existence of solutions for a short time, assuming the initial data is in a fourth-order Sobolev space and has Gaussian decay in the velocity variable (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass to every point in its domain. The resulting pointwise lower bounds have a sub-Gaussian rate of decay, which we show is optimal. The proof of mass-spreading is based on a stochastic process associated to the equation, and makes essential use of nonlocality. By combining this theorem with prior regularity results, we derive two important applications: $C^\infty$ smoothing in all three variables, even for initial data with vacuum regions, and a continuation criterion that states the solution can be extended for as long as the mass and energy densities stay bounded from above. This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.

math.AP↗

arXiv2017

Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.

math.AP↗

arXiv2016

Asymptotic stability for odd perturbations of the the stationary kink in the variable-speed $ϕ^4$ model

We consider the $ϕ^4$ model in one space dimension with propagation speeds that are small deviations from a constant function. In the constant-speed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and Muñoz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of non-constant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and Muñoz. This requires an understanding of the spectrum of the linearization around the variable-speed kink.

math.AP↗

arXiv2016

Linear stability of the Skyrmion

We give a rigorous proof for the linear stability of the Skyrmion. In addition, we provide new proofs for the existence of the Skyrmion and the GGMT bound.

math-ph↗

arXiv2015

An integro-differential equation without continuous solutions

We show an example of a non-symmetric integro-differential equation of order $α$, for $α\in (0,1)$, for which Hölder estimates do not hold even though the kernels are comparable to the fractional Laplacian.

math.AP↗

arXiv2015

Regularity and long-time behavior of nonlocal heat flows

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow approach to an optimal eigenvalue partition problem that, in light of the work of Caffarelli and Lin, is equivalent to a constrained harmonic mapping problem. In particular, we show that weak solutions of our heat equation converge as time approaches infinity to stationary solutions of this mapping problem. For these weak solutions, we also prove Lipschitz continuity and regularity of free interfaces, making use of a parabolic Almgren-type monotonicity formula.

math.AP↗

arXiv2011

Matrix probing: a randomized preconditioner for the wave-equation Hessian

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.

math.NA↗